On the functorial point of view in algebraic geometry.

Question

Here's a question I've been thinking about lately. I hope it's not too vague - I apologize in advance if this should be the case.

Suppose you want to do algebraic geometry using the $\textit{functorial}$ point of view.

I'm thinking of questions like $\textit{moduli problems}$. It seems to me that the functorial point of view should be very natural in this context - given that a moduli space is a representing object of a moduli functor by the very definition.

Now I always thought when doing moduli theory, one way to trying to study the moduli problem is by studying the $\textit{geometry}$ of the moduli space of the problem (assuming it exists etc...).

One way to study the geometry of a moduli space $\mathcal M$ is by studying its $S$-valued points $S \to \mathcal M$.

This feels like a very natural approach when it comes to moduli spaces, as one knows the set of morphisms $S \to \mathcal M$.

However, I feel like I'd be interested in morphisms having special properties, like (open/closed) embeddings, smooth morphisms etc.

Given that the construction of moduli schemes seems to be rather complicated most of the time I'd rather not like to go through the explicit construction when it comes to checking that a given morphism $S \to \mathcal M$ satisfies a certain property.

So ideally, I should be able to tell whether $S \to \mathcal M$ has certain properties $\textit{purely}$ from the $\textit{functorial point of view}$.

One example: Often one is able to compute the tangent space of a scheme $X$ as the set of $k[\epsilon]$-valued points of $X$ - something that one can understand rather explicitly for moduli spaces.

I think this is actually used to deduce that the tangent space of Hilbert schemes is given as certain first-order deformations. And that through computing these deformations one can really prove interesting results on the moduli problem "classified" by Hilbert schemes (i.e. by knowing the dimension of the tangent space etc).

So to summarize:

• Is there a (rather?) complete dictionary translating properties of morphisms of schemes into properties of their corresponding natural transformations?

I know that there are translation for open/closed embeddings. I'm not so sure about proper/smooth/unramified and other important properties though.

I'd be also interested in comments on whether my "guess" on how peoply try to work with moduli spaces is completely wrong or has a certain truth contained in it.

Answer 1

The theory of schemes as functors is not fully developed. I believe that many nice and elegant arguments, which could be made working purely with functors, are not invented yet, because the alternative theory already exists and people don't feel no need to redo the work. There are no textbooks about the functor of points which are as complete as e.g. Hartshorne or the stacks project. I think what might be useful though is a list of all the resources which I have been able to find over the last year.

• The algebraic groups book by Demazure & Gabriel develops the basics of scheme theory as functors. It is fairly well known, but it is also very old and the notation choices in the book are insane imo.

• The book about algebraic groups by Waterhouse does also define its schemes as functors. The notation and exposition are modern and pleasant, but the book is mostly about affine group schemes (= Hopf algebras) so not much (geometric) scheme theory is actually developed.

• There is the book SGA explained which describes the functor of points of the Hilbert-scheme if I remember correctly.

• The last chapter of Eisenbud's book is about the functor of points. Even though the definition of a scheme in that book is a locally ringed space, you can always take the description of the functor of points of a scheme as a definition in the alternative setting. Once you are over the initial hurdle (defining open and closed subfunctors and schemes etc.) everything flows smoothly and it isn't to hard to translate traditional results about schemes into the new language.

• There are unpublished lecture notes by Marc Nieper-Wißkirchen which develop schemes as functors from the start. They are the most complete reference I am aware of, but they are written in German. The only place in the internet where I can find them is at the end of this article by Martin Brandenburg. If the link doesn't work anymore, then you can ask me.

• There are lecture notes M392c: Algebraic Geometry by Arun Debray which introduce schemes as functors. But the notes are very incomplete. You can find them here. I hope the link does not die.

• Here is a very good tip. The authors Moerdijk and Reyes construct in their book Models for Smooth Infinitesimal Analysis well, models for synthetic differential geometry. They do this by constructing them as functors $C^\infty Ring\to Set$ on the category of $C^\infty Rings$. Their construction steps and theory is completely analogous to how you would define schemes as functors $Ring\to Set$. So by reading the beginning chapters of their book, and by doing everything in parallel with rings, you can learn a lot about the functor of points!

• Actually much of the theory of SDG can be interpreted in the big Zariski topos (where the functorial schemes live) (this excludes the order and integration axioms). So you can interpret huge portions of SDG as theorems about functorial schemes and Zariski local functors. But you will have to learn a little bit of categorical logic for that.

• Many SDG-books discuss somewhere in their appendix how $Zar$ is a model of SDG, so they also develop a little bit of its theory.

• The PhD thesis by Ingo Blechschmidt is a big dictionary between properties of schemes which be formulated through categorical logic, and the corresponding properties of traditional schemes. Since categorical logic can be interpreted in the category of Zariski local functors, this gives you in principle a dictionary between traditional scheme theory and functorial scheme theory. But there is definitely some translation work that has to be done, and learning categorical logic takes some time also.

• There is a text Formal Schemes and Formal Groups by Neil P. Strickland which contains some functorial scheme theory. But is far from a complete treatment.

• Sheaves and the functor of points is a book chapter by Michel Vaquie which might be interesting. But it is more high-level than the other references.

• A Master Course on algebraic stacks by Bertrand Toen develops functorial scheme theory.

• The fourth chapter of Michael Groechenig's Algebraic Stacks has something about functorial schemes.

• There is a text Elementary schematic geometry by Thiago Alexandre, but it is not very long and to general imo, if you are only interested in the special case $Sch \subset Sh(Aff,Zar)$.

I want to emphasize that using functorial schemes does not mean that you do not have access to the traditional theory and to the underlying locally ringed spaces. Once you have the basics, it is very easy to construct for every functorial scheme $S$ its little Zariski topos $Sh(S)$, its struture sheaf $\mathcal O_S$ and so on. All the traditional theorems apply. the only difference is, that you do not have to carry $\mathcal O_S$ around all the time. You only look at it when you need it.

Also: The functor of points becomes much cooler and simpler, when you also know a little bit of categorical logic. Learning categorical logic is not so hard as one might think. It is only important to not get lost in the depths of the type theory literature.